This work presents a method to produce conductive and auxetic composite membranes from a biobased and biodegradable matrix: polybutylene succinate (PBS). The conductivity was improved by the addition of graphene nanoplatelets (GNP) and the samples were produced via solution electrospinning. The membrane properties were shown to increase with increasing GNP concentration and the rotational speed of the collector. In particular, a membrane having 0.2% w/v GNP and fabricated at the highest collector speed (9.96 m/s) showed the highest electrical conductivity (1.56 × 10−4 S/m) while having a negative Poisson's ratio (NPR) of −1.5 in tension. To complete the analysis, mechanical characterizations showed that the presence of GNP led to a substantial increase in Young's modulus (234%) and tensile strength (190%) compared to the neat PBS membrane produced under the same conditions. Differential scanning calorimetry (DSC) revealed a slight crystallinity increase since GNP are acting as heterogeneous nucleating agents, while thermogravimetric analysis (TGA) showed an improved thermal stability for the GNP/PBS membranes. This unique combination of auxetic and conductive properties can be useful for a wide range of innovative applications such as electronic devices, smart textiles, biomaterials, and biomedical devices. Highlights Electrospinning was successful to produce polybutylene succinate (PBS) nanofibers. A careful control of the processing conditions led to auxetic fiber mats. Electrically conductive mats were produced by adding graphene nanoplatelets (GNP) to PBS. The PBS mechanical properties were highly improved (200%) with low GNP content (0.2%). PBS and GNP were mix in solution to produce electrically conductive nanofibers via electrospinning after optimization of the processing conditions.